L(s) = 1 | + (0.822 − 2.88i)3-s + 1.24i·5-s − 2.64·7-s + (−7.64 − 4.74i)9-s + 7.01i·11-s + 11.6·13-s + (3.58 + 1.02i)15-s − 4.52i·17-s + 16.2·19-s + (−2.17 + 7.63i)21-s − 25.5i·23-s + 23.4·25-s + (−19.9 + 18.1i)27-s + 9.49i·29-s − 28.7·31-s + ⋯ |
L(s) = 1 | + (0.274 − 0.961i)3-s + 0.248i·5-s − 0.377·7-s + (−0.849 − 0.527i)9-s + 0.637i·11-s + 0.895·13-s + (0.238 + 0.0681i)15-s − 0.266i·17-s + 0.854·19-s + (−0.103 + 0.363i)21-s − 1.11i·23-s + 0.938·25-s + (−0.740 + 0.672i)27-s + 0.327i·29-s − 0.926·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.274 + 0.961i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.274 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.805275676\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.805275676\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.822 + 2.88i)T \) |
| 7 | \( 1 + 2.64T \) |
good | 5 | \( 1 - 1.24iT - 25T^{2} \) |
| 11 | \( 1 - 7.01iT - 121T^{2} \) |
| 13 | \( 1 - 11.6T + 169T^{2} \) |
| 17 | \( 1 + 4.52iT - 289T^{2} \) |
| 19 | \( 1 - 16.2T + 361T^{2} \) |
| 23 | \( 1 + 25.5iT - 529T^{2} \) |
| 29 | \( 1 - 9.49iT - 841T^{2} \) |
| 31 | \( 1 + 28.7T + 961T^{2} \) |
| 37 | \( 1 - 33.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 67.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 24.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + 33.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 15.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 92.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 57.5T + 3.72e3T^{2} \) |
| 67 | \( 1 - 15.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + 70.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 76.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 127.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 74.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 127. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 23.1T + 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.002916386365172021474991160228, −8.414977237989782194547997175748, −7.36914473763435782595010215210, −6.87620320378066632596459078719, −6.06884535217847809692076602020, −5.09900293090204251417495373448, −3.75290016943174236755932470102, −2.87535308059901843845985028919, −1.81218406383563215828597147894, −0.55103391291074497158178166433,
1.18344849944422077704229639222, 2.86362493856909474514689165073, 3.55440930984679643674346690759, 4.46525961098624645862309398517, 5.51946391629441657036099319677, 6.08465868122670412874220340848, 7.35058579645837696184383364336, 8.287163312999174075255169844701, 8.920336273689992306150375969098, 9.619548899425987026643231459668